Rotation of spherical harmonics, 1996-2001 Recursive construction of general rotation matrices, avoiding the instabilities and inefficiencies of Wigner's Euler-angles-based formulas. Used for rotations of orbitals and multipoles. Currently used worldwide outside quantum chemistry in a variety of computer imaging and animation software. [165, 175, 176]

Orbital transformations and 2x2 Jacobi rotations

Resolution of general transformations into Jacobi rotation products, 1970-72 Resolution and reconstitution of non-singular N-dimensional matrices in terms of 2x2 rotations with a computational effort equal to that required for one matrix multiplication. Used for generating orbital transformations. [66, 82, 174]

Induction of configurational transformations by orbital transformations, 1999 Generation of the transformations that are induced in N-electron configuration spaces by transformations of orbitals [174]

Four-index transformations of electron repulsion integrals, 1993 Transformation of a set of electron-repulsions integrals by N(N-1)/2 sequential Jacobi rotations with a computational effort equal to that required for transformation by one NxN matrix. [152]

Electron interaction energies between atomic orbitals

Exponential-type orbitals, 1951-72 This first charge-density-driven vectorization, achieved by certain formal transformations, reduced the scaling of the computational effort for a molecular integral set from the fourth to the second power of the orbital space dimension. It has remained the method of choice for high-accuracy work on linear molecules in terms of exponential-type orbitals. [4, 27, 55; other integral contributions: 5, 8, 40, 41, 44, 49, 50, 51, 52, 80, 81]

Gaussian-type orbitals, 1970 An integral package for Gaussian orbitals (Biggmoli by R.Raffenetti in the ALIS code).

Determinant-based direct configuration interaction with general configuration selection, 2000-2001 A direct determinant-based configuration interaction method and fast code for large configuration spaces, applicable for full configuration spaces as well as for arbitrary configuration selections. Available in the GAMESS program system. [178]

Dynamic correlation energy extrapolation through intrinsic scaling, 2003-ongoing An extrapolation method (CEEIS) from double and triple excitations to full CI spaces that approximates the energies of large full configuration spaces within fractions of a millihartree by a computational effort reduced by many orders of magnitudes. The first practical full correlation energy recovery along reaction paths. [186 â€“189, 191, 198-200, 205]

General MCSCF approach based on super-CI expansion, 1976-81 An early well-converging multi-configuration self-consistent field method and code based on orbital optimization by iterative improvements of natural orbitals through single-excitation annihilations. Popular with researchers experimenting with nonstandard CI expansions because of its capability to deal with arbitrary configuration selections. [109, 114, 116, 120]

General MCSCF approach based on Jacobi-type orbital rotation sequences, 2003 An MCSCF method for large arbitrary configuration spaces based on orbital optimizations of full rather than second order energy expressions. Available in the GAMESS program system. [181]

Stable MCSCF method for excited states, 2003 An MCSCF method for excited states in the presence of lower states of like symmetry, based on a functional that prevents root-flipping related instabilities. [181]

A priori generation of full valence space wavefunctions free of configurational deadwood, 2000-ongoing It is shown that only the determinants of a very small subspace contribute to the correlation recovered by the full configurational valence space when appropriate localized orbitals are used. A systematic method for the effective a-priori determination of these correlating subspaces is developed, leading to a drastic shortening of the CI expansions in large molecules. [178, 179, 202, 203]

Semi-empirical recovery of dynamic correlation

Intra-atomic correlation correction in the FORS model, 1985 The FORS model, in its atom-localized form, proved perfectly suited for an ab-initio quantification of the Atoms-in-Molecules Correlation-Correction concepts, advanced by Moffit and Hurley's in the fifties and sixties to obtain an estimate of dynamic electron correlation contributions. [128, 129]

A dynamic correlation energy estimate based on localized orbital, 2002 A very simple formula for estimating dynamic correlation energies in terms of electron pair populations within and between the dominating localized orbitals yields correlation energies with a mean deviation of 2-3 kcal/mol for a large number of SCF dominated molecules. [180]

Total energy and orbital energies in SCF theory, 1977 The SCF energy was shown to be approximately equal to 1.5 times the sum of the occupied orbital energies. Found particular resonance among density functional theorists. [110]

A physical interpretation of groundstate energy shifts is derived from the variation principle. The variational analysis shows covalent bond formation in H2+ to be the result of structural changes in the kinetic energy functional that are caused by the orbital delocalization due to electron sharing. They induce a firmer attachment of electrons to the nuclei, which accounts for the potential energy lowering identified by the virial theorem. Adopted in books by Mulliken, Fukui, Kutzelnigg and others. By 1986, one of the 100 most cited papers of the Reviews of Modern Physics. [22, 62, 69, 74, 108, 193, 194, 204]

The model of molecules formed by bonded atoms, the functional concept derived from two hundred years of experimental chemistry, differs from the physical theory, which is based on electrons and nuclei. The derivation of an ab-initio quantification of the former from the latter, which would enhance the interaction between chemical experimentalists and quantum-chemical theorists, is still an on-going enterprise.

Exact expansion of exact self-consistent-field orbitals in terms of quasi-atomic orbitals, 2003 It is shown that the exact SCF (or DFT) density matrix can be exactly expanded in terms of a unique intrinsic minimal basis set of quasiatomic orbitals that deviate only slightly from free-atom orbitals. Also yields zeroth-order approximations to the most important correlation orbitals. Application to Bloch orbitals in crystals extracts quasi-atomic orbitals from plane-wave representations. [183, 184, 185, 190]

Exact expansion of full valence-space MCSCF orbitals in terms of quasi-atomic orbitals, 1976-2003 The full valence space MCSCF molecular orbitals are shown to be intrinsically expressible in terms of basis-set-independent molecular orbitals that have quasiatomic character. The many electron wavefunction is expressed in terms of quasi-atomic states. Quantitative measures for electron migration and charge transfer are obtained. [109, 122, 123, 124, 196, 197]

Chemical orientation of quasiatomic orbitals, 2003 An intrinsic criterion and algorithm is found that determines intra-atomic orthogonal transformations of the quasiatomic orbitals on all atoms in a molecule such that each of the resulting oriented quasiatomic hybrid orbitals is involved in at most one bond. [196, 197]

Gradient fields, gradient extremals and reaction channels, 1986-94 Analyses of steepest descent fields show that intrinsic reaction paths, i.e. steepest descents from transition states to reactant and product minima, do not necessarily have the character of reaction channels. They do so only when running close to gradient extremals. The latter are new types of curves on potential energy surfaces whose properties are analyzed and for which a following-algorithm is developed. Also used for global transition state searches. [134, 153, 161]

Critical Points, 1993-94 The quadratic steepest descent method is shown to have advantages for minimum searches as well as transition state searches. Also, approximate transition states are predicted analytically using only the hessians from reactant and product minima. [158, 163, 164]

Reaction path bifurcations, 1986 A split into several branches complicates the identification of reaction paths. An analysis, achieved by introducing the valley-ridge inflection points, yields a classification of bifurcation patterns according to their location relative to transition states. [135]

Intersections between potential energy surfaces of like symmetry

Violation of the weak non-crossing rule, 1990-1997 According to the weak non-crossing rule, conical intersections between one-configuration-dominated closed-shell singlet states were considered extremely unlikely since they are impossible between single determinants, in particular when involving ground states. The analysis of ozone and its valence-isoelectronic analogues showed that such intersections do occur when the adiabatic states are superpositions of diabatic states that embody significant internal non-dynamic correlations varying markedly as the diabatic states switch dominance in the adiabatic states. [145, 150, 154, 171]

Intersections of intersection seams, 1997-1999 The analysis of the ozone case showed that intersection seams can consist of several branches that can cross at lower-dimensional nodes, so that the global intersection seam forms a topologically multi-connected network. This notably occurs when two crossing states having the same overall irreducible representation belong to different irreducible representations in a coordinate subspace of higher symmetry. [167, 173]

Surface patterns near conical intersections, 1991 The surface topographies in intersection regions are shown to be either of the sloped or the peaked type and are further classified by the characteristics of their singular gradient field patterns. Became a popular introduction to conical intersections. [151]

Quantum chemical geometric phase theorem, 1990-1999 A new derivation of the phase theorem without explicit reference to intersections. Entails a generalization to higher order intersections. Also developed was the use of the phase theorem for the ab-initio determination of conical intersections. [145, 150, 173]

Diabatic States, 1993-1999 A new ab-initio determination of diabatic states from adiabatic states based on enforcing configurational uniformity in the electronic wavefunction over the relevant region of the potential energy surface. [155, 170]

Long-range interaction between atoms, 2007-ongoing

It is shown how long-range interactions between bonded atoms can be resolved into multipole and dispersion interactions by means of resolving the configuration interaction wavefunction in terms of a zeroth-order component and a dynamic correlation component. [44, 201, 205]

Accurate ab-initio dissociation energies of C2, N2, O2, F2, 2005 Using the CEEIS method, extrapolating to a complete basis set and including relativistic effects, the dissociation energies of C2, N2, O2, F2 are determined within 0.6 kcal/mol of the experimental values. [189]

Accurate ab-initio potential energy curve of F2, 2007-ongoing Using the CEEIS method, extrapolating to a complete basis set and including relativistic effects, an accurate potential energy curve of F2 is determined, which yields the entire vibrational spectrum with a mean absolute deviation of 5 cm-1. The long-range part is shown to be the superposition of quadrupole repulsions and dispersion attractions. [198, 199, 200, 205]

Triatomic potential energy surfaces

Isomerization of nitrosyl-hydride, HNO NOH, 1977 Determination and atomic interpretation of the isomerization curves and barriers for the lowest singlet and triplet states by full-valence-space MCSCF calculations. [114 and Ph.D. thesis, M.G. Dombek, Iowa State Univ, 1977)]

Global and local features of the lowest two singlet states of ozone, 1990-97 Isomerization O3 (obtuse angled) <â€”> O3 (equilateral ring). Conical intersection seam. The ground state and the lowest excited state of the same symmetry are superpositions of the closed-shell s14p4 and s12p6 diabatic states, the former dominating at the open minima, the latter dominating at the ring-minimum of the ground state. The diabatic configurations switch dominance in the adiabatic wavefunctions near the transition states from the ring to the open structures of the ground state, and the excited state's global minima occur nearby. Within 0.04 Ã… of this region, the two states are degenerate, as proven by quantitative application of the phase theorem. This was the first instance of a conical intersection between two closed-shell singlets of like symmetry. The one-dimensional intersection seam was found to consist of four loops connected at three nodes. The global determination of both surfaces in the three-dimensional internal coordinate space shows them being close to each other over an extended region, containing the intersection seam, favoring radiationless transitions. Dissociation from the open ground state minimum into O2+O occurs by an end atom moving away with little change in the apex angle and a small barrier. [145, 150, 154, 166, 167, 168]

Conical intersections in the molecules O3, S3, SO2, S2O, 1997 All members of this valence-isoelectronic series are found to possess the same kind of conical intersection between the 1A1 ground state and the lowest excited 1A1 state as found in ozone. [171]

Enthalpy of formation of water, 2006 The enthalpy of formation of water is recovered within 0.03 kc/mol. The energy of atomization is recovered within 0.06 kcal/mol. [191]

Concerted dihydrogen exchange between ethylene and ethane, 1982â€¨H3C-CH3 + H2C=CH2H2C=CH2 + H3C-CH3Exhibits limits of the Woodward-Hoffmann rules by showing that this paradigm of a symmetry-allowed reaction has in fact a barrier of about 70 Kcal/mole. [125]

Ring opening of cyclopropylidene to allene, 1986-91â€¨H2(CCC)ring H2 H2C=C=CH2, The first global potential energy surface in terms of three primary internal coordinates, with twelve additional spectator coordinates optimized at each point in the primary coordinate space. The calculated 8 Kcal/mol barrier agrees with experimental inferences. The ring opening is not stereo-specific. The transition state maintains the reactant's C2v symmetry. The reaction path bifurcates after the transition state near an intersection with an excited state, yielding two stereo-isomeric lower symmetry products. On the downhill paths, the hydrogen pairs can perform free, but synchronized internal rotations. Analogous reactions of substituted reactants are stereo-specific due to non-bonded steric and long-range electrostatic interactions between the substituents. [136, 146, 147, 148, 149]